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Mirrors > Home > MPE Home > Th. List > psgnevpmb | Structured version Visualization version GIF version |
Description: A class is an even permutation if it is a permutation with sign 1. (Contributed by SO, 9-Jul-2018.) |
Ref | Expression |
---|---|
evpmss.s | ⊢ 𝑆 = (SymGrp‘𝐷) |
evpmss.p | ⊢ 𝑃 = (Base‘𝑆) |
psgnevpmb.n | ⊢ 𝑁 = (pmSgn‘𝐷) |
Ref | Expression |
---|---|
psgnevpmb | ⊢ (𝐷 ∈ Fin → (𝐹 ∈ (pmEven‘𝐷) ↔ (𝐹 ∈ 𝑃 ∧ (𝑁‘𝐹) = 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3510 | . . . 4 ⊢ (𝐷 ∈ Fin → 𝐷 ∈ V) | |
2 | fveq2 6663 | . . . . . . . 8 ⊢ (𝑑 = 𝐷 → (pmSgn‘𝑑) = (pmSgn‘𝐷)) | |
3 | psgnevpmb.n | . . . . . . . 8 ⊢ 𝑁 = (pmSgn‘𝐷) | |
4 | 2, 3 | syl6eqr 2871 | . . . . . . 7 ⊢ (𝑑 = 𝐷 → (pmSgn‘𝑑) = 𝑁) |
5 | 4 | cnveqd 5739 | . . . . . 6 ⊢ (𝑑 = 𝐷 → ◡(pmSgn‘𝑑) = ◡𝑁) |
6 | 5 | imaeq1d 5921 | . . . . 5 ⊢ (𝑑 = 𝐷 → (◡(pmSgn‘𝑑) “ {1}) = (◡𝑁 “ {1})) |
7 | df-evpm 18549 | . . . . 5 ⊢ pmEven = (𝑑 ∈ V ↦ (◡(pmSgn‘𝑑) “ {1})) | |
8 | 3 | fvexi 6677 | . . . . . . 7 ⊢ 𝑁 ∈ V |
9 | 8 | cnvex 7619 | . . . . . 6 ⊢ ◡𝑁 ∈ V |
10 | 9 | imaex 7610 | . . . . 5 ⊢ (◡𝑁 “ {1}) ∈ V |
11 | 6, 7, 10 | fvmpt 6761 | . . . 4 ⊢ (𝐷 ∈ V → (pmEven‘𝐷) = (◡𝑁 “ {1})) |
12 | 1, 11 | syl 17 | . . 3 ⊢ (𝐷 ∈ Fin → (pmEven‘𝐷) = (◡𝑁 “ {1})) |
13 | 12 | eleq2d 2895 | . 2 ⊢ (𝐷 ∈ Fin → (𝐹 ∈ (pmEven‘𝐷) ↔ 𝐹 ∈ (◡𝑁 “ {1}))) |
14 | evpmss.s | . . . . 5 ⊢ 𝑆 = (SymGrp‘𝐷) | |
15 | eqid 2818 | . . . . 5 ⊢ ((mulGrp‘ℂfld) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1}) | |
16 | 14, 3, 15 | psgnghm2 20653 | . . . 4 ⊢ (𝐷 ∈ Fin → 𝑁 ∈ (𝑆 GrpHom ((mulGrp‘ℂfld) ↾s {1, -1}))) |
17 | evpmss.p | . . . . 5 ⊢ 𝑃 = (Base‘𝑆) | |
18 | eqid 2818 | . . . . 5 ⊢ (Base‘((mulGrp‘ℂfld) ↾s {1, -1})) = (Base‘((mulGrp‘ℂfld) ↾s {1, -1})) | |
19 | 17, 18 | ghmf 18300 | . . . 4 ⊢ (𝑁 ∈ (𝑆 GrpHom ((mulGrp‘ℂfld) ↾s {1, -1})) → 𝑁:𝑃⟶(Base‘((mulGrp‘ℂfld) ↾s {1, -1}))) |
20 | 16, 19 | syl 17 | . . 3 ⊢ (𝐷 ∈ Fin → 𝑁:𝑃⟶(Base‘((mulGrp‘ℂfld) ↾s {1, -1}))) |
21 | ffn 6507 | . . 3 ⊢ (𝑁:𝑃⟶(Base‘((mulGrp‘ℂfld) ↾s {1, -1})) → 𝑁 Fn 𝑃) | |
22 | fniniseg 6822 | . . 3 ⊢ (𝑁 Fn 𝑃 → (𝐹 ∈ (◡𝑁 “ {1}) ↔ (𝐹 ∈ 𝑃 ∧ (𝑁‘𝐹) = 1))) | |
23 | 20, 21, 22 | 3syl 18 | . 2 ⊢ (𝐷 ∈ Fin → (𝐹 ∈ (◡𝑁 “ {1}) ↔ (𝐹 ∈ 𝑃 ∧ (𝑁‘𝐹) = 1))) |
24 | 13, 23 | bitrd 280 | 1 ⊢ (𝐷 ∈ Fin → (𝐹 ∈ (pmEven‘𝐷) ↔ (𝐹 ∈ 𝑃 ∧ (𝑁‘𝐹) = 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 {csn 4557 {cpr 4559 ◡ccnv 5547 “ cima 5551 Fn wfn 6343 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 Fincfn 8497 1c1 10526 -cneg 10859 Basecbs 16471 ↾s cress 16472 GrpHom cghm 18293 SymGrpcsymg 18433 pmSgncpsgn 18546 pmEvencevpm 18547 mulGrpcmgp 19168 ℂfldccnfld 20473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-xor 1496 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-ot 4566 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-xnn0 11956 df-z 11970 df-dec 12087 df-uz 12232 df-rp 12378 df-fz 12881 df-fzo 13022 df-seq 13358 df-exp 13418 df-hash 13679 df-word 13850 df-lsw 13903 df-concat 13911 df-s1 13938 df-substr 13991 df-pfx 14021 df-splice 14100 df-reverse 14109 df-s2 14198 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-0g 16703 df-gsum 16704 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mhm 17944 df-submnd 17945 df-grp 18044 df-minusg 18045 df-subg 18214 df-ghm 18294 df-gim 18337 df-oppg 18412 df-symg 18434 df-pmtr 18499 df-psgn 18548 df-evpm 18549 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-cring 19229 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-dvr 19362 df-drng 19433 df-cnfld 20474 |
This theorem is referenced by: psgnodpm 20660 psgnevpm 20661 evpmodpmf1o 20668 mdet0pr 21129 odpmco 30657 evpmid 30717 cyc3evpm 30719 |
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